Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio

We study the finite horizonMerton portfolio optimization problem in a general local-stochastic volatility setting. Using model coefficient expansion techniques, we derive approximations for the both the value function and the optimal investment strategy. We also analyze the ‘implied Sharpe ratio’ and derive a series approximation for this quantity. The zeroth-order approximation of the value function and optimal investment strategy correspond to those obtained by Merton (1969) when the risky asset follows a geometric Brownian motion. The first-order correction of the value function can, for general utility functions, be expressed as a differential operator acting on the zeroth-order term. For power utility functions, higher order terms can also be computed as a differential operator acting on the zeroth-order term. While our approximations are derived formally, we give a rigorous accuracy bound for the higher order approximations in this case in pure stochastic volatility models. A number of examples are provided in order to demonstrate numerically the accuracy of our approximations.